There also exists a generalisation of Pearl's schema to quantum mechanics which replaces classical causes with quantum ones. Various versions of this approach exist[1]}, [2]}, [3]}, but here we will follow the approach of ref [4]} which takes `causal relations to be relations of influence between quantum nodes mediated by unitary evolution.' In this approach, a quantum causal model is defined on a set of quantum nodes - each node is a pair of Hilbert spaces, one associated with an incoming system, and one associated with an outgoing system, and the idea is that an agent could potentially perform an operation \(O_i\)  at a node \(A_i\) , thus implementing a transformation on the incoming system to produce the outgoing system. Thus a quantum causal model constitutes a directed acyclic graph on a set of quantum nodes, together with a set of quantum channels \(\lbrace  \rho _{A_i | Pa(A_i)} \rbrace \)  such that the incoming system for node \(A_i\)  is given by the output of the channel \(\rho _{A_i | Pa(A_i)}\)  acting on the set \(Pa(A_i)\)  of its parents in the DAG, and every channel \(\rho _{A_i | Pa(A_i)}\)  commutes with every other channel \( \rho _{A_j | Pa(A_j)}\) . The process as a whole can be described by a `process operator' which maps the set of operations \(O_i\)  implemented by observers at nodes to an operation which represents the net effect of performing all of these operations in the context of the particular process; usually it is assumed that the channels should obey a `quantum Markov condition,' which requires that the process operator can be written as the product of each of the individual channels \(\rho _{A_i | Pa(A_i)}\) . It can be seen that this approach has much in common with the process matrix framework - the `nodes' play the role of laboratories where agents freely choose their operations, and the `process operator' plays a similar role to the process matrix. However, the emphasis here is on defining a specific causal structure rather than allowing indefinite causal orders.
